Verified Answer :

  • Distribute the negative signs: \(-2 + 2 + 8i + 2 + 6i\).
  • Group the real components: \(-2 + 2 + 2 = 2\).
  • Group the imaginary components: \(8i + 6i = 14i\).
  • Combine the results to get the final simplified expression: \(\boxed{2 + 14i}\).

Explanation

  1. Analysis of the expression

The given expression is \(-2 - (-2 - 8i) - (-2 - 6i)\). To simplify this, we need to distribute the negative signs through the parentheses and then combine the real and imaginary parts separately. The expression consists of real components and complex components involving the imaginary unit \(i\). Recall that the distributive property states that \(-(a + b) = -a - b\), so we can apply this to remove the parentheses.

  1. Distributing the negative signs

Applying the distributive property, the expression expands as follows:
\[-2 - (-2 - 8i) - (-2 - 6i) = -2 + 2 + 8i + 2 + 6i.\]
Here, we have distributed the negative sign to each term inside the parentheses:

  • \(-(-2) = 2\)
  • \(-(-8i) = 8i\)
  • \(-(-2) = 2\)
  • \(-(-6i) = 6i\)

Combining these, we get:
\[-2 + 2 + 2 + 8i + 6i.\]

  1. Combining real and imaginary parts and final result

Now, group the real and imaginary parts to simplify further:

  • Real parts: \(-2 + 2 + 2 = 2\)
  • Imaginary parts: \(8i + 6i = 14i\)

Adding these together, the final simplified expression is \(2 + 14i\). Comparing this result with the given choices, we find it corresponds to option A.

Examples

Complex numbers are widely used in electrical engineering to analyze AC circuits. For instance, impedance, which describes how much an electrical component opposes the flow of alternating current, is often represented as a complex number \(Z = R + jX\), where \(R\) is resistance and \(X\) is reactance. By simplifying complex expressions, engineers can easily calculate total circuit behavior, such as total impedance in series circuits or voltage drops across components, much like simplifying \(-2 - (-2 - 8i) - (-2 - 6i)\) into \(2 + 14i\).

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